Two-point boundary value problem for systems of pseudo-differential equations under boundary conditions containing pseudo-differential operators

Abstract

This paper deals with a two-point boundary value problem for pseudodifferential equations and for systems of second order pseudodifferential equations under boundary conditions containing pseudodifferential operators. The need to consider pseudodifferential operators is caused by two reasons, first, such equations appear more and more often in applied problems, and second, by considering such equations, it is possible to achieve the well-posedness of the boundary value problem in the Schwartz space S and in its dual space.First, we consider a scalar pseudodifferential equation with a symbol belonging to the space $C_{-\infty}^{\infty}$, consists of infinitely differentiable functions of polinomial growth. For this equation it is found of the boundary condition under which a specific type the boundary value problem is well-posed in the space S. In addition, an example of a differential-difference equation and a specific boundary condition with a convolution-type pseudo-differential operator under which this boundary value problem is well-posed in the space S are given.Then we consider a system of two pseudodifferential equations with symbols from the space $C _ {-\infty} ^ {\infty}$. For this system, we prove the existence of a well posed boundary value problem in the space S. The Fourier transform and the reduction of the system to a triangular form are used in the proof. In this case, we also give an example of a system and a specific boundary condition under which this boundary value problem is correct in the space S.Thus, the work proves that for any pseudo-differential equation, as well as for a system of two pseudo-differential equations, there is always a correct boundary value problem in the $S$ space, while the boundary conditions contain pseudo-differential operators. The algorithm for constructing correct boundary conditions is also indicated. They are pseudo-differential operators whose symbols depend on the symbols of pseudo-differential equations.